Convection-enhanced diffusion in a critical case

We consider a diffusion process with a random time-independent and spatially stationary drift. The two-dimensional case is scaling-wise critical; we focus on a divergence-free drift, which can be written as the curl of the Gaussian free field.

In the presence of an unavoidable small-scale cut-off, we prove that the process is borderline super-diffusive: Its annealed second moments grow like $t\sqrt{\ln t}$.

This refines a recent result of Cannizzaro, Haunschmid-Sibitz and Toninelli; the method however is completely different and appeals to quantitative stochastic homogenization of the generator that can be reformulated as a divergence-form second-order elliptic operator.

In fact, it embeds homogenization techniques into a renormalization group argument reminiscent of the heuristics in the physics literature.

This is joint work with Georgiana Chatzigeorgiou, Peter Morfe, and Lihan Wang.